492 



HYDRODYNAMICS. 



Discharge 

 of Fluids 



from 

 Orifices. 



PROP. XII. 



< * > "~Y~ 1 "" / To determine the horizontal distance to which water 

 will be projected from orifices in the side of a vessel, 

 and the nature of the curve which it will describe. 



Pi ATE Let A BCD, Fig. 5. be a vessel of water, which is 



cccxvm. discharged at O through the bent tube GEO, in the 

 F 'S- ' direction OP. If the water were influenced by no other 

 force but that with which it is projected, it would move 

 uniformly in the direction m o P, with a velocity equal 

 to that which a heavy body would acquire in falling 

 through the height QO. But as it is acted upon by 

 gravity, as soon as it escapes from the orifice O it will ob- 

 viously describe some curve line O n p. Make the ele- 

 mentary space O m=m o, and OP=2 OQ. Draw PM 

 parallel to ON, and join QM. Let fall from the points 

 m, o, P, the vertical lines m n, o p, PV, which will be 

 parallel to OM, and complete the parallelograms O m n R, 

 O op S, OPVT. Let us now suppose, that in the time 

 in which the water would have described the space 

 O m, the force of gravity would have caused it to fall 

 through the height OR; and that in the time in 

 which it would have described the space O o, it would 

 have descended through OS by the force of gravity 

 alone. Now, since the fluid at O is solicited by two 

 forces, one of which, viz. the force of projection, 

 would carry it through the space O m in a certain 

 time, while the other, viz. the force of gravity, would 

 carry it through the space OR in the same time, the 

 fluid will at the end of the given time be found at n. In 

 like manner it may be shewn, that at the end of the 

 time in which the water would have described O o uni- 

 formly, it will be found in the point p. But since O m, 

 O o represent the times in which the water reaches the 

 points m, p of its path, and since in these times the force 

 of gravity has caused the water to fall through the spaces 

 mn, op, then as the spaces are proportional to the 

 squares of the times, we have m n, opO m 1 : O o 1 , that 

 is on account of O m n R, Oop S, being parallelograms 

 OR : OS=R n' : S p 2 , which is the relation between the 

 abscissas and the ordinates of the Apollonian hyperbola. 

 (See CONIC SECTIONS, Sect. IV. Prop. XII. Cor.) 



It would be unnecessary to proceed any farther in 

 explaining and demonstrating the geometrical construc- 

 tion which is usually given for finding the amplitude 

 either of oblique or horizontal jets, as the construc- 

 tion and the demonstration of it are exactly the same as 

 that which we have given in our article GUNNERY, Vol. 

 X. p. 572, &c. for the parabolic path of projectiles. 

 The two classes of phenomena, and the mathema- 

 tical laws by which they are regulated, are exactly the 

 same. 



PROP XIII Discharge 



op> AI11< of Kluids 



To determine the pressure exerted on the interior of from 

 conduit pipes by the water which they convey. 



Orifices. 



Let the fluid column, Plate CCCXVIII. Fig. 5. No. 2. PLATE 

 be divided into an infinite number of equal and verti- rccxviiu 

 cal laminae GF gf. Then if we abstract friction, it is Fig. 5, 

 obvious that all the points of the same lamina have the No - 2 - 

 same velocity, and that this velocity is the same in all 

 the laminae. Ifqr represent the section of the con- 

 tracted vein at the orifice p n, the velocity of the la- 

 minae is to the velocity in q r, as the area of the orifice 

 q r is to the area of the section GF ; for at every instant 

 there passes out of q r a small prism of water equal to 

 GF gf, and therefore these prisms have velocities reci- 

 procally proportional to their bases. (See Prop. I. p, 

 488. ) If we therefore call h the constant height of wa- 

 ter in the reservoir, D the diameter of the tube, d that 

 of the orifice q r, and if we consider that the velocity in 

 qr is that due to the height h, and may be expressed by 



t/fi, then D- : d 2 =r\/A : ri<r> tne velocity of the wa- 

 ter in the pipe. But as the velocity t^h is due to the 

 height h, the velocity ^ will be due to the altitude 



-=rj . But since each particle of fluid that reaches the 

 extremity PN of the pipe tends to move with the velo- 

 city \/'h, while it moves only with the velocity =^ ,. 



every point of P p or N n upon which it rests must be 

 pressed with a force equal to the difference of the pres- 

 sure due to the velocities */h, and ^ , that is, every 



part of the pipe will be pressed with a force equal to 

 d'k 



Cor. 1. If an aperture very small in relation to each 

 of the orifices PN, pn is made in the side of the pipe, 

 the water will issue with a velocity due to the height 



h =r^. This height will vanish when d = D, or 



when the whole aperture PN is opened. 



See Bossut's Traite D'Hydrodi/namique, Tom. II, 

 chap. xi. p. 197, &c. from which the preceding proposi- 

 tion is taken. 



SCHOLIUM. 



In page 513, 51 4, of the present article, will be found, 

 a set of valuable experiments by Bossut, in which he 

 has measured the quantity of water discharged by aper- 

 tures in the side of the pipe. The agreement between 

 the formula and the observed results, is very striking.* 



* In a scries of very recent and interesting experiments on the discharge of liquids through small orifices, made by M. Hachette, 

 of which some account will be found in the following Chapter, he has discovered that the quantity of fluid discharged by orifices va- 

 ries by placing an obstacle at some distance from the orifice. Daniel Bernoulli made an experiment on this subject, and concluded 

 from it, that an obstacle does not alter the quantity of fluid discharged. In his experiment, however, the time of the flow was too 

 short for obtaining correct results. 



M. Hachette employed a circular orifice, 20 millimetres in diameter, which discharged water from a large vessel into a vessel pla- 

 ced at a great distance from the orifice. The surface of the water in the vessel sank about six decimetres in 10' 21". The plane 

 face of an obstacle was presented at different distances from the orifice, and the jet fell perpendicularly on this plane. The follow- 

 ing were the results : 



Distance of the Obstacle in Millimetres. 



128 80 50 24 4 



Corresponding Times in which the Surface of the Water sunk Six Decimetres. 

 10' 21" 10' 25" 10' 26" 11' 13" 15' 54" 



Hence it follows, that at the distance of 128 millimetres (5.039 inches), the obstacle produces no effect; but that, at the distance of 

 Ceur millimetres 0.157 of an inch, the time is increased rather more than one-half. 



